Uniform estimates of Landau-de Gennes minimizers in the vanishing elasticity limit with line defects

For the Landau-de Gennes functional modeling nematic liquid crystals in dimension three, we prove that, if the energy is bounded by $C(\log\frac{1}{\varepsilon}+1)$, then the sequence of minimizers $\{\mathbf{Q}_{\varepsilon}\}_{\varepsilon\in (0,1)}$ is relatively compact in $W_{\operatorname{loc}}^{1,p}$ for every $1<p<2$.

This extends the classical compactness theorem of Bourgain-Brézis-Mironescu [Publ.

Math., IHÉS, 99:1-115, 2004] for complex Ginzburg-Landau minimizers to the $\mathbb R\mathbf P^2$-valued Landau-de Gennes setting.

Moreover, We obtain local bounds on the integral of the bulk energy potential that are uniform in $ \varepsilon $, improving the estimate that follows directly from the assumption.